The terrestrial and lunar reference frame in lunar laser ranging

Transcription

1 Journal of Geodesy (1999) 73: 125±129 The terrestrial and lunar reference frame in lunar laser ranging C. Huang, W. Jin, H. Xu Shanghai Astronomical Observatory, Academia Sinica, 80 Nandan Road, Shanghai , People's Republic of China Received: 21 August 1996 / Accepted: 17 November 1998 Abstract. The initial value problem and the stability of solution in the determination of the coordinates of three observing stations and four retro-re ectors by lunar laser ranging are discussed. Practical iterative computations show that the station coordinates can be converged to about 1 cm, but there will be a slight discrepancy of the longitudinal components computed by various analysis centers or in di erent years. There are several factors, one of which is the shift of the right ascension of the Moon, caused by the orientation deviation of the adopted lunar ephemeris, which can make the longitudinal components of all observing stations rotate together along the longitudinal direction with same angle. Additionally, the frame of selenocentric coordinates is stable, but a variation or adustment of lunar third-degree gravitational coe cients will cause a simultaneous shift along the re ectors' longitudes or rotation around the Y axis. Key words. Terrestrial reference frame á Lunar reference frame á Lunar laser ranging 1 Introduction The lunar laser ranging (LLR) station located at McDonald Observatory received successfully the rst photon from the retro-re ector on the Moon in August At that time, there was only one observatory which was working on LLR, and its precision of ranging was at sub-meter level. In the past quarter of a century, the technique has been improved continually, and two stations, located at Correspondence to: C. Huang clhuang@center.shao.ac.cn; Tel.: /6191; Fax: /4618 Grasse (France) and Haleakala (USA), have also taken part in LLR observation, since 1982 and 1984 respectively. The 2.7-m telescope of McDonald was replaced by a 76-cm one in More than normal points of range measurements have been accumulated to date. The ranging precision has been improved to 2 or 3 cm at the single range between the Moon and the Earth, and it is expected to be improved to millimeter level by the end of this century (IERS 1991). The primary scienti c motive of LLR is to verify Einstein's equivalence principle; however, LLR has also been used to determine many dynamic and kinematic parameters of the Earth, such as the Earth orientation parameter (EOP), the coordinates of stations, the Earth's elastic parameters, precession and nutation, and a series of lunar parameters related to the orbit and libration of the Moon (Xu et al. 1996), and the selenocentric coordinates of retro-re ectors (Dickey et al. 1994). The precision of the terrestrial coordinates of stations determined by LLR analysis centers is about several centimeters. Generally speaking, it is internal precision; the results provided by various analysis centers and obtained in di erent years by one analysis center di er much from each other. And in the case of the lunar coordinates of retro-re ectors, precision is much worse than that of the station coordinates. The precision and stability of the terrestrial and lunar reference frame determined by LLR are discussed in this paper. 2 LLR data reduction The observation equation of LLR at rst order is Dq ˆ q o q c ˆ X 8 9 oq >: >;DK R n e ok 1 where q o observed range between the telescope and the retrore ector

2 126 q c theoretical range calculated by models partial derivative of the range to parameter DK correction to initial parameter K e accidental error R n truncation error induced by development at rst order P P q ;DK i DK R n ˆ 2 2! Generally, the computation is decomposed into two stages. First, corrections of more than 50 parameters are obtained from least-square adustment of the whole available measurements, and post- t residuals are also obtained, these parameters include geocentric coordinates of observatories, precession, nutation, the orbital elements of the Moon around the Earth and some lunar parameters such as third-degree potential harmonics, Love number, fractional moment of inertia di erences, rotation dissipation and selenocentric coordinates of re ectors. In the second stage, the universal time (UT) and the variation of latitude at observatory, D/, are solved from the post- t residuals at a single observatory. The observation equation can be written as Dq i ˆ >: >;DUT 9 >: >;D/ X 8 9 @K 3 It is also in rst approximation. The data span used in the second step is short (from 1 to 5 days), so the last term in Eq. (3) can be treated as a constant. 3 The terrestrial and lunar reference frames The terrestrial and lunar reference frames are constituted by the geocentric coordinates of three LLR stations and the selenocentric coordinates of four retro-re ectors for LLR. In the processing, the parameters to be solved are the corrections to their initial values. This means that the range q is developed around these initial values as a Taylor expansion of rst order, and the value R n, whose order is higher than one, is taken as minterm. In order to avoid the e ect induced by the inappropriateness of the initial value, we can usually develop the function to higher order or iterate several times. However, if developed at higher order, the observation equation becomes nonlinear, making the adustment di cult, so it is usual to adopt the multiple iteration method. As one part of this paper, the results of the terrestrial coordinates of three stations and the selenocentric coordinates of four retro-re ectors are discussed using the method of multiple iteration. All the measurements between the three stations and four retro-re ectors during the period January 1988 to December 1995 are used to make least-square adustment. The adopted lunar/ planetary ephemeris and lunar libration are DE403/ LE403 of the Jet Propulsion Laboratory (JPL). 3.1 The selenocentric coordinates of four retro-re ectors The iterative results of the selenocentric coordinates of four retro-re ectors are listed in Table 1 (left-hand side). The initial values are taken from Huang et al. (1996), Table 1. The iterative results of re ectors' selenocentric cylindrical coordinates in the PA system S 31 and S 33 adusted S 31 and S 33 not adusted r (m) k ( ) Z (m) r (m) k ( ) Z (m) Re ector No. 0 A B C D Re ector No. 2 A ) ) ) ) B ) ) ) ) C ) ) ) ) D ) ) ) ) Re ector No. 3 A B C D Re ector No. 4 A B C D A: initial value; B: adustment result; C: one iteration; D: two iterations

3 127 where the selenocentric cylindrical coordinates (r, k, z) are referred to the principal axis (PA) system (Eckhardt 1981). Here the third-degree lunar potential harmonics, including S 31 and S 33, are adusted in this solution. We can see from the left-hand side of Table 1 that the iterative results of the r and Z components of all four re ectors uctuate at the level of several centimeters, while all longitudinal components k vary around their mean value with about a similar, large uctuation, )0.500 (a variation of 1.0 in longitude equals a variation of about 8.4 m in lunar equator), after the rst adustment. In addition, all k of the four re ectors are changed at the second iteration with the same values, which are and )1.047, respectively. This means that the frame constructed by four re ectors is stable in the reduction, but its k axis orientation will probably wobble with a certain angle. It is well known that the re ector coordinates and integrated physical librations are calculated in the PA system, and due to the existence of third- and higherdegree lunar potential harmonics, their torques do not average to zero and produce a rotation between the PA system and the mean Earth/rotation (ME) system. Usually, this rotation is composed of two parts; the rst part is a constant term, and the second is a partial derivative with respect to gravitational harmonics changes (Ferrari et al. 1980; Eckhardt 1981). When S 31 and S 33 are adusted in the solution, the second term mentioned above causes the same shift in the longitudinal direction of the re ectors (Eckhardt 1973; Williams et al. 1996; Williams, pers. commun. 1997). Based on the above, xing S 31 and S 33, the iterative results of selenocentric cylindrical coordinates for four re ectors in the PA system are also listed in Table 1 (right-hand side). It is clear that the large shift of the longitudinal component no longer exists, but a similiar rotation around the Y axis appears, )0.30 and in the last two iterations, respectively. This is caused by the fact that the third-degree harmonics C 30 and C 32 are also adusted. Practical calculation shows that all the shifts of longitude or rotation angles around the Y axis will vanish after xing C 30, C 32, S 31 and S 33. Therefore, it is con rmed that the frame of selenocentric coordinates is stable, but a variation or adustment of lunar third-degree gravitational coe cients will cause a similar rotation. 3.2 The geocentric coordinates of three telescopes The determination of the station coordinates is much better than that of re ectors. The iterative results of the terrestrial coordinates of three stations are listed in Table 2; their reference epoch is converted to by the plate motion model of NNR-NUVEL 1, and the initial values are also adopted from Huang et al. (1996). It is shown in Table 2 that the coordinates of three stations can converge quickly after only one iteration, and the extent of convergence is better than 1 cm. However, the station coordinates determined by various analysis centers will di er from each other. And Table 2. The iterative results of geocentric cylindrical coordinates for three stations because di erent models are used and the observational normal points used are di erent each year, the results of coordinates from the same analysis center will also vary each year, especially in the longitudinal component. As an example, the results of the geocentric coordinates of three telescopes determined by JPL every year and their adopted lunar/planetary ephemeris, collected and converted from published papers, are listed in Table 3; these values are derived using the formulation of the relativistic solar-system barycentric reference frame and have been converted to the reference epoch of by the corresponding plate motion model. It can be found easily from Table 3 that a whole rotation along the longitudinal direction for the three stations between 1990 and 1991 exists, it is about )37 milliarcsecond or )1.1 m on the equator. This phenomenon also exists between SSC(JPL)95M01 (DE247/ LE247 is used) and SSC(JPL)95M02 (DE402/LE402 is used). The reason for this is discussed in the following. The lunar time angle H can be expressed in an observation equation as (Williams and Melbourne 1982) H ˆ a M a G UT 1 k r (m) k ( ) Z (m) MLRS (new site) Initial value Adustment result Iteration result Haleakala Tmtr Initial value Adustment result Iteration result CERGA Initial value Adustment result Iteration result where a M right ascension of the Moon a G UT 1 Greenwich Sidereal Time k geocentric longitude of stations H is a constant at a certain measurement; the error of a M or a G will a ect the result of k. Generally, the station coordinates take part in the prior adustment as global parameters. IERS Bulletin series or other series are adopted as the initial value of UT1; their accuracies are about or better than 0.2 ms, i.e. 3 mas, in these years (Muller, pers. commun. 1997), the error of the initial UT1 will come into k directly via Eq. (4). The right ascension of the Moon, a M, is given by the adopted lunar ephemeris in the reduction. Because the adopted position of the vernal equinox and precession constant are di erent, a systematic deviation of orientation among ephemerides will exist (Folkner et al. 1994). For example, JPL has published ephemeris series 0**, 1**, 2**, 3** and the present series 4**. DE303/

4 128 Table 3. The geocentric cylindrical coordinates of three stations in SSC(JPL)**M** Year r (m) k ( ) Z (m) Lunar/planet. ephemeris MLRS (new site) 89M DE121/LE65 90M DE303/LE303 91M DE229/LE229 92M DE246/LE246 93M DE210/LE210 94M DE247/LE247 95M DE247/LE247 95M DE402/LE402 96M DE328/LE328 Haleakala Tmtr 89M DE121/LE65 90M DE303/LE303 91M DE229/LE229 92M DE246/LE246 93M DE210/LE210 94M DE247/LE247 95M DE247/LE247 95M DE402/LE402 96M DE328/LE328 CERGA 89M DE121/LE65 90M DE303/LE303 91M DE229/LE229 92M DE246/LE246 93M DE210/LE210 94M DE247/LE247 95M DE247/LE247 95M DE402/LE402 96M DE328/LE328 LE303 is used in 1990, and then DE2**/LE2** are used from 1991 to For DE200 and several ephemerides after, their zero points of right ascension are kept xed; later they are moved in the solutions, and for DE403, the zero point of right ascension is aligned with the IERS celestial frame. Therefore, the reference frames of these ephemerides are not xed, because their zero points are allowed to move away; as a result, there will be a systematic shift in orientation. DE310 and DE245 are compared here as an example. Two hundred geocentric spheric coordinates of selenocenter (R, a, d) between MJD47762 (23 August 1989) and MJD48757 (14 May 1992), with an interval of 5 days, are obtained from DE310 and DE245, respectively. Their di erence is the systematic shift between two ephemerides. DR uctuates between 0.35 and 0.42 m with a mean value of 0.39 m, while Dd uctuates between )2.69 and 2.68 milliarcsecond with a mean value of 0.10 milliarcsecond. And Da, the systematic shift in orientation between DE310 and DE245, uctuates between )17.24 and )15.86 milliarcsecond with a mean value of )16.51 milliarcsecond and is plotted in Fig. 1. So, it is reasonable that there is a systematic shift in the geocentric longitudinal direction of telescopes presented by JPL between 1990 and Besides the reason that the observational normal points used are di erent each year, there are another two factors: the uncertainty of the adopted initial UT1 series in adustment, and the systematic shift of the right ascension of the Moon, a M, due to di erent lunar ephemerides. 4 Summary Fig. 1. The discrepancy of right ascension of the Moon between DE245 and DE310 between MJD47762 and MJD48757 From the analyses mentioned above, the following conclusions can be drawn. 1. Stepwise iteration can be used to weaken the e ect of the inappropriateness of initial parameters in the determination of coordinates of stations and re ectors by LLR measurements. For telescopes, they can be converged quickly at about 1-cm level after one or two iterations.

5 Iterative computations show that, for retro-re ectors, their frame is stable, but a variation or adustment of lunar third-degree gravitational coe cients will cause a similar rotation. 3. The systematic orientation shift of the adopted lunar ephemeris will make the geocentric coordinates of all telescopes rotate together along the longitudinal direction with an angle which cannot be ignored. Acknowledgments. The authors acknowledge all the people who have been undertaking the daily maintenance of the LLR observation and the regular data storage during the long period of continuous registrations, especially under the present poor funding circumstances. The geodetic group of JPL is thanked for their kindness in providing lunar and planetary ephemerides. The authors are also indebted to Dr. Jim Williams, JPL, for his careful review of this paper and for the many udicious suggestions which contributed valuable ideas for clarifying particular sections of this paper. References Dickey JO, Bender PL, Faller JE, Newhall XX, Ricklefs RL, Ries JG, Shelus PJ, Veillet C, Whipple AL, Wiant JR, Williams JG, Yoder CF (1994) Lunar laser ranging: a continuous legacy of the Apollo program. Science 265: 482±490 Eckhardt DH (1973) Physical librations due to the third and fourth degree hamonics of the lunar gravity potential. Moon 6: 127±134 Eckhardt DH (1981) Theory of the libration of the Moon. Moon Planets 25: 3±49 Ferrari AJ, Sinclair WS, Siogren WL, Williams JG, Yoder CF (1980) Geophysical parameters of the Earth±Moon system. J Geophys Res 85 (B7): 3939±3951 Folkner WM, Charlot P, Finger MH, Williams JG, Sovers OJ, Newhall XX, Standish EM (1994) Determination of the extragalactic-planetary frame tie from oint analysis of VLBI and LLR measurement. Astron Astrophys 287: 279±289 Huang CL, Jin WJ, Xu HG (1996) The LLR reference frame. Annals of Shanghai Observatory. Acad Sin 17: 169± International Earth Rotation Service (IERS) Annual Report, Observatoire de Paris, piv11 Williams JG, Melbourne WG (1981) In: Calame O (ed) Highprecision Earth rotation and Earth±Moon dynamics. Reidel Dordrecht, p 313 Williams JG, Melbourne WG (1982) Comments on the e ect of adopting new precession and equinox corrections. In: Calame O (ed) Proc. IAU Coll. 63, High precision Earth rotation and Earth-Moon dynamics: Lunar distances and related observations. Reidel Dordrecht, 293±303 Williams JG, Newhall XX, Dickey JO (1996) Lunar moments, tides, orientation, and coordinate frames. Planet Space Sci 44: 1077±1080 Xu HG, Jin WJ, Huang CL (1996) The secular acceleration of the moon determined from lunar laser ranging data. Earth, Moon Planets 73: 101±106